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# Learning to Sample Hard Instances for Graph Algorithms

; Proceedings of The Eleventh Asian Conference on Machine Learning, PMLR 101:503-518, 2019.

#### Abstract

\textit{Hard instances}, which require a long time for a specific algorithm to solve, help (1) analyze the algorithm for accelerating it and (2) build a good benchmark for evaluating the performance of algorithms. There exist several efforts for automatic generation of hard instances. For example, evolutionary algorithms have been utilized to generate hard instances. However, they generate only finite number of hard instances. The merit of such methods is limited because it is difficult to extract meaningful patterns from small number of instances. We seek for a probabilistic generator of hard instances. Once the generative distribution of hard instances is obtained, we can sample a variety of hard instances to build a benchmark, and we can extract meaningful patterns of hard instances from sampled instances. The existing methods for modeling the hard instance distribution rely on parameters or rules that are found by domain experts; however, they are specific to the problem. Hence, it is challenging to model the distribution for general cases. In this paper, we focus on graph problems. We propose \textsc{HiSampler}, the hard instance sampler, to model the hard instance distribution of graph algorithms. \textsc{HiSampler} makes it possible to obtain the distribution of hard instances without hand-engineered features. To the best of our knowledge, this is the first method to learn the distribution of hard instances using machine learning. Through experiments, we demonstrate that our proposed method can generate instances that are a few to several orders of magnitude harder than the random-based approach in many settings. In particular, our method outperforms rule-based algorithms in the 3-coloring problem.